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International Journal of Number Theory
c World Scientific Publishing Company

Self-points on elliptic curves of prime conductor

Christophe Delaunay
Université de Lyon
Université Lyon1
CNRS, UMR 5208 Institut Camille Jordan
Batiment du Doyen Jean Braconnier
43 blvd du 11 novembre 1918
F - 69622 Villeurbanne Cedex
Francedelaunay@math.univ-lyon1.fr
Christian Wüthrich
CSAG, Section de mathématiquesÉcole polytechnique fédérale
1015 Lausanne
Switzerland
christian.wuthrich@epfl.ch

Received (Day Month Year)
Revised (Day Month Year)
Let E be an elliptic curve of conductor p. Given a cyclic subgroup C of order p in E[p],we construct a modular point PC on E, called self-point, as the image of (E, C) on X0 (p)under the modular parametrisation X0 (p) E. We prove that the point is of infinite
order in the Mordell-Weil group of E over the field of definition of C. One can deduce alower bound on the growth of the rank of the Mordell-Weil group in its PGL 2 (Zp )-tower
inside Q(E[p∞ ]).
Mathematics Subject Classification 2000: 11G05, 11G18, 11G40
Keywords: Elliptic curves, modular curves

1. Introduction
Let p be a prime number and let E/Q be an elliptic curve of conductor p. It is known GL2 (Fp ) is surjective. (Wethat the Galois representation ρ̄p : Gal(Q(E[p])/Q)
will recall the classification of curve of prime conductor in Section 2.) Let ψ be an
isogeny E
 E 0 of degree p with kernel C ⊂ E[p]. The field Q(C) of definitionof ψ is a degree p + 1 extension in Q(E[p]) fixed by a Borel subgroup in GL2 (Fp ).
We consider now the point xC = (E, C) on the modular curve X0 (p). Its image
under the modular parametrisation ϕE : X0 (p) E will be denoted by PC . It isa point on E defined over Q(C). Since the point represents the curve on itself, we
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Christophe Delaunay, Christian Wuthrich

will call this PC a self-point on E. Let K be the Galois closure of Q(C), it is the
PGL2 (Fp )-extension inside Q(E[p]).

Theorem 1. The self-point PC is of infinite order in E(Q(C)). As C runs throughall cyclic subgroups of order p in E[p], the set {PC }C generates a subgroup of rank
p in E(K) on which PGL2 (Fp ) acts like the Steinberg representation.

The Steinberg representation is a p-dimensional irreducible representation of
PGL2 (Fp ) whose definition will be recalled in Lemma 5.
There is a possibility to construct higher self-points PD for all cyclic subgroups
D of order pn in E[pn ]. If C is the unique order p subgroup in D, let Ĉ be its dual in E/D.
(E/C)[p]. Denote by C (D) the image of Ĉ under the natural map E/C
Then PD is the image of (E/D, C (D) ) under the modular parametrisation ϕE . We
will prove in Theorem 8 that all the points PD are of infinite order. The points PD
generate a group of rank pn + pn−1 − 1 in E(Kn ) where Kn is the PGL2 (Z/pn Z)extension in Q(E[pn ]).
In Section 7, we will compute the parity of the p-Selmer group of E over Q(C)and the root number of E over this field using methods and results of Shuter [Shu06],
Rohrlich [Roh06] and Dokchitser [Dok05]. These computations lead naturally to theconjecture that the rank of E(Q(C)) is even if and only if p is congruent to 1 modulo
4. Interestingly our self-points do not behave like Heegner points with respect toroot numbers.
In Section 8, we will give some numerical examples. We compute explicitly the
self-points on the curves of conductor 11, 17 and 19. Eventhough they are fairly
simple to define and have rather small canonical height, their coordinates havesurprisingly complicated expressions. It seems plausible that the self-points are notdivisible by any integer in E(Q(C)).
These self-points have been considered earlier by Harris in [Har79] where heobtained asymptotic formula for the rank of E(Kn ) of the form mentioned above.
The investigation is tightly linked to non-commutative Iwasawa theory. Though
in the form presented here, it is concerned with the Iwasawa theory of the p-adic
Lie extension K∞ /K for the prime p of multiplicative reduction. These are thefirst known points of infinite order defined over the GL2 (Zp )-extension Q(E[p∞ ])naturally attached to E. Nevertheless the computations of Rohrlich on the root
numbers for the irreducible Artin representations of GL2 (Zp ) and the recent results
for the p-Selmer group in [CFKS06] suggest that the rank of E(Q(E[pn ])) should
grows much faster than pn , at least when p ≡ 3 (mod 4).
In a forthcoming paper [Wut07], we will consider self-points on elliptic curves
whose conductors are not necessarily prime. In this general setting, the situation ismuch more involved.

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Self-points on elliptic curves of prime conductor

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2. Classification of curves of prime conductor
The first few examples of elliptic curves E/Q of prime conductor p are listed in
the following table. Only the strong Weil curves, i.e. optimal curves with respect tothe modular parametrisation ϕE : X0 (p) E are included. (“s” stands for split
multiplicative reduction at p and “n” for non-split multiplicative reduction.)
Curve
TorsionReduction
Rank

11a1
5
s
0

17a1
4
s
0

19a1
3
s
0

37a1
1
n
1

37b1
3
s
0

43a1
1
n
1

53a1
1
n
1

61a1
1
n
1

67a1
1
s
0

73a1
2
s
0

We summarise in the following proposition the facts known about curves ofprime conductor.
Proposition 2. Let E be an elliptic curve of prime conductor p and let E 0 be thestrong Weil curve in its isogeny class. Then the Tamagawa number cp = − ordp (j)of E0 at p is equal to the order of the torsion subgroup of E(Q) and its Manin
constant is trivial. We are in one of the following three cases• The curve E0 has no torsion points defined over Q. Then the `-adic Galois
representations ρ` : Gal(Q̄/Q) Aut(T` (E)) = GL2 (Z` ) is surjective
for all `, hence there are no isogenies on E0 = E defined over Q. The
Tamagawa number cp at p is 1.
• We have #E0 (Q)tors = 2. Then the prime p is of the form u2 + 64 for some
integer u ≡ 3 (mod 4). The curve E is one of the two isogenous curves.
2
E0 : y 2 + xy = x3 − u+1
4 x + 4x − u2
E1 : y 2 + xy = x3 − u+1
4 x − xFor these curves we have that E(Q) = Z/2Z and X(E/Q)[2] = 0. All ρ `for ` 6= 2 are surjective.
• We have #E0 (Q)tors > 2. Then E0 will be among the curves of conductor
11, 17, 19 or 37 shown in the table above.
The curves in the second case are treated in [Set75] and [Neu71] and [Neu73].
They are called Neumann-Setzer curves in [SW04]. The first few primes for which
we are in this situation are 73, 89, 113, 233, 353, 593, . . .
Proof. According to [MO89], the curves of conductor p with a torsion point
have been classified by Miyawaki [Miy73] and Setzer in [Set75]. In Corollaire 5.2
in [MO89], it is shown that E(Q)tors is isomorphic to the group of connected components of the Néron model of E at p. Furthermore, we know by Serre [Ser72] thatfor a semi-stable curve either there is a curve with a rational non-trivial torsion
point in its isogeny class or the curve does not admit any isogeny defined over Q.
Moreover [Ser96] shows that the representation ρ` is surjective for all primes ` forFebruary 22, 2007 9:36 WSPC/INSTRUCTION FILE

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Christophe Delaunay, Christian Wuthricha semi-stable curve unless the curve admits an isogeny of degree ` defined over
Q. The statement about the Manin constant is proved in [AU96]. It remains tocite [SW04] for the result on the Mordell-Weil group and the 2-torsion part of theTate-Shafarevich group of the Neumann-Setzer curves.
Note that in particular the representation ρp is surjective. Given a cyclic subgroup C of E[p], then the field Q(C) is the field fixed by a Borel subgroup and istherefore a non-Galois degree p + 1 extension without any proper sub-extensions.
As C runs through all possible cyclic subgroups in E of degree p, we obtain p + 1
such extensions and their compositum is the field K fixed by the centre of GL2 (Fp ).
The Galois group of K/Q is therefore G ∼= PGL2 (Fp ).
3. The main theorem
The aim of this section is to prove Theorem 1. The theorem is split up in two
propositions that we will prove separately.
Proposition 3. The self-points PC are of infinite order.
Proof. Recall that K was defined to be the Galois closure of Q(C). Note that the
points {PC }C form a single orbit under the action of Gal(K/Q) in E(K), because
G ∼= PGL2 (Fp ) acts transitively on the set of all C, which we will identify fromnow on with P1 (Fp ). It is therefore enough to show that one of the self-points is of
infinite order.
First we fix an embedding of Q̄ into Q̄p . We consider the modular parametrisation over Qp . The modular curve X0 (p) over Q̄p has a neighbourhood of the cusp ∞Z
consisting of couples (A, C) of a Tate curve of the form A = Q̄×
p /q together withth
its subgroup generated by the p root of unity. The parameter q is a p-adic analytic
uniformiser at ∞, so that the Spf Qp [[q]] is the formal completion of X0 (p)/Qp at
the cusp ∞, see [DR73].
P
Let fE = an q n be the normalised newform associated to E and so fE /q · dq isthe associated differential, and since we know that the Manin constant is trivial, it
coincides with ϕ∗E (ωE ) where ωE is the invariant differential on E. The rigid analytic
map induced by ϕE on the completion can now be characterised as
logE (ϕE (q)) =Z ϕE (q)
O

ωE =

Z q
0

fE

X an
dq
=
· qn .
qnn>1Here logE denotes the formal logarithm associated to E from the formal groupÊ(pZp ) to Ĝa (pZp ) = pZp .
Since E has multiplicative reduction at p, there is exactly one of the xC in thisneighbourhood, we call it x0 . The other self-points are p-adically close to the second
cusp 0. Write P0 for ϕE (x0 ). Let qE be the p-adic Tate parameter associated to E.

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By the above characterisation of ϕE , we know now that
X an
a2 2
logE (P0 ) =· qEn = qE +
q +··· .
n
2 E
n>1Since the valuation of qE is equal to the Tamagawa number cp > 1, the sum on theright converges and its value will be congruent to qE modulo p2·cp . In particular thevalue is non zero and hence is a non torsion element in pcp Zp . Therefore P0 is a
non-torsion point in Ê(pcp Zp ).
This second proposition will now end the proof of Theorem 1.
Proposition 4. The points {PC }C generate a group of rank p in E(K). The only
relation is that the sum of all PC is equal to the image ϕE (0) of the cusp 0 on X0 (p),which is a torsion point in E(Q).
Proof. We prove first the relation between the self-points. By definition the sum
P
C PC is the image of the class of the divisor
X
D=((xC ) − (∞))
C∈P1 (Fp ) E where J0 (p) is the Jacobian of X0 (p). We compare
under the map J0 (p)this divisor to the divisor of the function j − j(E) on X0 (p). We haveX
div(j − j(E)) = −p · (0) − (∞) +(xC )C∈P1 (Fp )= D − p · ((0) − (∞)) .
Hence the sum of all PC is equal to p · ϕE (0). If p > 37, then ϕE (0) is a point oforder at most 2 and p is an odd prime, so we have p · ϕE (0) = ϕE (0). For p 6 37,
we also have p ≡ 1 modulo the order of E(Q)tors .
Further we need the following lemma defining the Steinberg representation.
L
Lemma 5. Let V be the Q[G]-module
C∈P1 (Fp ) Q · eC with the natural action ofG∼
the sum of two irreducible Q[G]-modules,= PGL2 (Fp ) on P1 (Fp ). It splits intoP
the first 1-dimensional generated by C eC and the second p-dimensional given byonX
XaC = 0 ,St =aC · eCcalled the Steinberg representation of G.

P
Proof. It is clear that V splits as W0 ⊕ St with W0 generated by C eC . We start
to prove the irreducibility of St by showing that there are no one-dimensional subP
modules. Let v =aC · eC be a fixed vector. By subtracting the obvious fixedP
vectoreC , we may assume that one of the coordinates in v is zero. But then v
can not be fixed by G since the action of G on P1 (Fp ) is transitive.

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Let now g be an element of order p in G. The eigenvalues of g acting on St
are exactly the p-th roots of unity. If now the Q[G]-module St splits into two submodules, then necessarily one of them would have to be one-dimensional, which wehave just excluded above. Therefore St is irreducible.
Now we can end the proof of the proposition. There is a G-equivariant map fromV to E(K) ⊗ Q given by sending eC to PC . Since the sum of all PC is torsion, the E(K) ⊗ Q is an injective
space W0 is in the kernel. The restricted map St
morphism of Q[G]-modules by Proposition 3. Hence the group generated by the
self-points {PC } is a full lattice in the image of this map which is a p-dimensional
vector space.
Corollary 6. The point PC is not divisible by pcp in E(Q(C)).
Proof. This can be deduced from the proof of Proposition 3. It is shown there thatthe p-adic valuation of logE (PC ) in Qp is equal to cp . Let p be the place of Q(C)corresponding to the chosen embedding Q̄ in Q̄p , then Q(C)p = Qp . Since the orderof the group of components of E over Qp and the number of non-singular points in
the reduction E(Fp ) are both prime to p, the question whether PC is divisible by
pcp in E(Qp ) is the same as to ask whether it is divisible by it in the formal groupÊ(pZp ). But now the statement about the valuation of logE (PC ) shows that PC can
not be divided by pcp in E(Qp ).
3.1. More on the Steinberg representation
The character χSt of the irreducible representation ρSt : G
 Aut(St) of G ∼=PGL2 (Fp ) can be explicitly described in the following manner. Of course χSt (1) = p.
There are three different types of non-trivial conjugacy classes in G. First the classof elements of order p, where χSt takes the value 0. The second type consists ofconjugacy classes containing a diagonal matrix. For all such classes, the value of χ St
is +1. Finally χSt takes the value −1 on the remaining classes. This representationand its twist by the only non-trivial representation of G of dimension 1 are called
Steinberg representations in [Lan02, page 712] and [Sil70].
4. The torsion subgroup of E over K
The group PGL2 (Fp ) admits only one non-trivial 1-dimensional representation χ(over C). It is defined asχ : PGL2 (Fp )det Fp× /Fp×2

 {±1}

with kernel PSL2 (Fp ). Hence there is a unique quadratic subfield L inside K and it
has to be the unique quadratic extension that is only ramified at p, that is√p−1
L = Q( p∗ ) with p∗ = (−1) 2 · p .

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Theorem 7. Let E be an elliptic curve of prime conductor p and let K, L and
Q(C) as before. Then we have E(K)tors = E(L)tors . Moreover, we have E(K)tors =E(Q)tors except for the Neumann-Setzer curves E1 and the curves 17a2, 17a3 and17a4. In particular E(Q(C))tors = E(Q)tors .
Proof. First let ` be a prime for which the mod ` representationρ̄` : Gal(Q̄/Q) Aut(E[`]) is surjective. Suppose that E(K) contains an `torsion point. Then Q(E[`]) must be contained in K since K/Q is Galois. Since
Gal(K/L) = PSL2 (Fp ) is a simple group, either Q(E[`]) is equal to K or to L. But
since the degree [Q(E[`]) : Q] is ` · (` − 1)2 · (` + 1) it is certainly impossible that
Q(E[`]) = L. But it can not be equal to K either as the equation` · (` − 1)2 · (` + 1) = p · (p − 1) · (p + 1)
implies that ` < p, unless ` = p = 2 which is impossible because p > 11. But if` < p then the left hand side is not divisible by p unless ` = 2 and p = 3 which is
not possible either. Hence E(K) does not contain any `-torsion points. For the firstclass of curves in Proposition 2 this means that E(K)tors = 0.
So we may suppose that E has a torsion point over Q. By the classification, weknow that there is a unique prime ` 6 5 dividing #E(Q)tors . The above guarantees
that E(K)tors is an `-primary group.
Suppose now that E(K) has a torsion point S which is not defined over L.
By taking a multiple of S, we may suppose that ` · S is defined over L. Hencethe Galois closure of L(S) is of degree at most `! 6 120 as the Galois group is apermutation group of the set {T |` · T = ` · S}. This is a contradiction with thefact that PSL2 (Fp ) is a simple group with at least # PSL2 (F11 ) = 660 elements.
Therefore E(K)tors = E(L)tors .
Next we compute E(L)tors = E(K)tors for the four isogeny classes of curves with
#E(Q)tors > 2 this can be done directly and the result is given in the table below.
11a1 11a2 11a3 17a1 17a2 17a3 17a4 19a1 19a2 19a3 37b1 37b2 37b3
Curve
E(Q)tors
51
5
4 [2,2] 2
4
3
1
3
3
1
3
E(K)tors 51
5
4 [2,4] [2,2] [2,4] 3
1
3
3
1
3
Finally, we treat the Neumann-Setzer curves. We have to compute the 2-primary√
torsion of E over L. For such curves we have L = Q( p). For the curve E0 , we finduthat the 2-torsion points are generated by T = ( 4 , − u8 ) and (2 · i, −i) with i2 = −1.
So we have E(L)[2] = Z/2Z T . Trying to divide T by 2, we find that the S suchthat 2 · S = T are the conjugates of α2 α α2 , −with α4 − p u α2 + 16 p2 = 0 .
S=
2p 4
2p
√
√
Note that α2 = p2 · (u + p) belongs to Q( p), but its norm is equal to −16 · p2√
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Christophe Delaunay, Christian Wuthrichover L and we conclude that
E0 (K)tors = E0 (L)tors = E0 (Q)tors = Z/2Z T .
For E1 we find that a basis of the 2-torsion point can be given by u + √p u + √ p 
T1 = (0, 0)
and
T2 =
,−
.
8
16
So E[2] ⊂ E(L). The x-coordinates of 4-torsion points are defined over Q(i) or overthe field defined by β 2 − 2 x(T2 ) β − 1. The discriminant of this equation is equal√
to 18 (p + u p) whose norm is p. So we conclude that there are no 4-torsion points
defined over L and that
E0 (K)tors = E0 (L)tors = Z/2Z T1 ⊕ Z/2Z T2E0 (Q)tors = Z/2Z T1 .

This end the proof of the proposition.
5. Curves with isogenies
Let E be an elliptic curve of conductor p who admits an isogeny defined over Q.
According to the Proposition 2, the prime p is either of the form u2 + 64 and there
are two curves in the isogeny class or p is in the set {11, 17, 19, 37} as shown in thetable in Section 2.
Let E 0 /Q be one of the elliptic curves isogenous to E. The image of a given cyclic
subgroup C of order p on E under the isogeny is a cyclic subgroup C 0 of order p in
E 0 , since the degree of the isogeny is prime to p. Similar to the self-points, we canalso consider the point x0C = (E 0 , C 0 ) on the modular curve X0 (p) and its image PC0
in E. This point, just as the self-point PC is also defined over Q(C). The proof of
Proposition 3 applies just the same to show that the point PC0 is of infinite order inE(Q(C)). The set of points {PC0 } as C runs through all cyclic subgroups C of order
p in E will be a copy of the Steinberg representation in E(K), hence of rank p.
The only way that this group generated by PC0 could intersect the group generated
by PC is that, for every given C, the points PC and PC0 are linearly dependant.
Unfortunately we are not able to show that PC and PC0 are linearly independent,
but it might well be the case.
So it looks likely that we actually have rank(E(K)) > i · p where i is the numberof curves in the isogeny class of E over Q. For the curves of conductor 11, 17 and 19
we will prove it later. Note that if all the 2-torsion points of E were defined over Q– something which can never happen for optimal curves of prime conductor – thenwe would have a relation between the points {PC0 }. For more details on these more
subtle questions, we refer to [Wut07].
6. Higher Self-points
Let E/Q be an elliptic curve of prime conductor p. We also fix a cyclic subgroup
C of order p on E. Applying the Atkin-Lehner involution wp to xC gives a point

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Self-points on elliptic curves of prime conductor9yC = (E/C, Ĉ) on X0 (p) whose image in E(Q(C)) differs from ap · PC by a rational
2-torsion point. Recall that ap = ±1 with the sign depending on whether E has
split or non-split multiplicative reduction at p. The cyclic group Ĉ of order p on
the isogenous curve E/C is the kernel of the dual isogeny.
Let n > 1 be an integer and let D be a cyclic subgroup of E of order pn whichcontains C. The isogeny ψD of kernel D factors through
ψD

E
ψ



 E/D

C

ψ D/
E/C

We consider the point yD = (E/D, ψD/C (Ĉ)) on X0 (p). Note that this is well-definedsince ψD/C (Ĉ), which we denoted by C (D) in the introduction, is indeed cyclic oforder p. We define PD to be the image of yD in E(Q(D)). The Galois closure of
Q(D) is denoted by Kn , it is a PGL2 (Z/pn Z)-extension of Q inside Q(E[pn ]).
Theorem 8. The group generated by the points PD in E(Kn ) as D runs throughall the cyclic subgroups of order pk with k 6 n is of rank pn + pn−1 − 1. In particular
all of the points PD are of infinite order.
Proof. Let B be the subgroup of D of order pn−1 and let C be the subgroup oforder p. In fact, we will prove the following trace relation
trQ(D)/Q(B) (PD ) = ap · PB .

(6.1)

The Hecke operator Tp acts on the divisor (yB ) − (∞) as X(E/B)/A, (ψB/C (Ĉ) + A)/A −(∞)
Tp (yB ) − (∞) =A

where A runs over all cyclic subgroups of order p in E/B with trivial intersection
with ψB/C (Ĉ). Let D0 be the preimage of A under ψB . It has order pn and the factthat it intersects trivially with ψB/C (Ĉ) proves that D 0 is a cyclic subgroup of E.
Moreover the cyclic subgroup (ψB/C (Ĉ) + A)/A of order p is equal to ψD0 /C (Ĉ).
Hence

 X
 X
Tp (yB ) − (∞) =(E/D0 , ψD0 /C (Ĉ)) − (∞) =(yD0 ) − (∞)
D0

D0with the sum this time running over all cyclic subgroups D of E of order pn containing B. Now we consider the image of the above under the modular parametrisation
ϕE : J0 (p) E. The left hand side will map to ap · PB and the right hand side toX
ap · PB =PD0 = trQ(D)/Q(B) (PD )0D0since the Galois conjugates of yD are exactly all of the yD0 .

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Having proved the formula (6.1), we know by induction on n that the points
PD are of infinite order. Let us look at the rational PGL2 (Z/pn Z)-representation V
whose basis {eD } is in bijection with the set P1 (Z/pn Z) of all cyclic subgroups D
in E of order pn . According to [Sil70], page 58, this representations decomposes as
V = W0 ⊕ W1 ⊕ W2 ⊕ · · · ⊕ W n
where Wi is an irreducible representation of dimension pi − pi−2 if i > 1, denoted
u1,i in [Sil70], and W1 = St is the Steinberg representation considered earlier. Thereis a Gal(Kn /Q)-equivariant map from V to E(Kn ) ⊗ Q sending eD to PD . We proveby induction on n that the kernel of this map is the trivial subspace W0 . For n = 1
this coincides with the statement of Theorem 1. To prove the statement it is now
enough to note that if the map from Wn to E(Kn ) ⊗ Q were not injective then allof the PD would have to be of finite order in contradiction with (6.1).
Hence rank(E(Kn )) > dimQ (V ) − dimQ (W0 ) = (p + 1) · pn−1 − 1.
A little bit stronger, we can even claim that we have the following boundrank(E(Kn )) > pn + pn−1 − 1 + rank(E(Q)) .
Such asymptotic lower bound were already found by Harris in [Har79]. It is
likely that one could obtain the formula i · (pn + pn−1 − 1) with i begin the numberof curves in the isogeny class of E over Q. On the other hand this bound is far
from the upper bounds coming from non-commutative Iwasawa theory. We find inlemma 3.3 in [HS05] a lemma due to Howson giving an upper bound of the form
C · p3n for some constant C. Moreover the computations of root numbers (as in thenext section) suggests that the rank of E(Kn ) should grow much faster than our
bound. See also the recent work of Coates, Fukaya, Kato and Sujatha [CFKS06].
But we wish to emphasise that the self-points are so far the only explicitly known
points of infinite order in the tower E(Kn ).
Of course, one can more generally consider higher self-points. Let ` be any prime
different from p. Given a cyclic subgroup D of order ` in E[`], one can consider the
points ϕE (E/D, (C + D)/D) which are defined over Q(C, D). They satisfy tracerelations like (6.1). These points live in the PGL2 (Z` )-extension of Q(C). We will
investigate these points in more details and greater generality in [Wut07].
7. Root numbers and Parity of the Selmer GroupWe return now to the first layer of the tower. Let E be an elliptic curve of conductor
p and let K and Q(C) be the fields as before. Since the point PC is of infinite
order the L-series of E over K and Q(C) should vanish at s = 1; at least if we
believe the conjecture of Birch and Swinnerton-Dyer. More precisely the L-function
L(E, ρSt , s) twisted by the Steinberg representation which appeared in the proof of
Propositions 4 should vanish at s = 1. Unfortunately, it is not even known whether
or not this L-series admits an analytic continuation to s = 1. One is often able toFebruary 22, 2007 9:36 WSPC/INSTRUCTION FILEjsp

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11

predict the vanishing of the L-series just by computing the associated root number,i.e. the sign of the conjectured functional equation. But unlike in the case of Heegner
points, there will not be a link between the non-triviality of the self-point and someroot number.
We define the root number of a representation as the product of the local factors as in [Dok05]. It should coincide with the root number appearing in thefunctional equation of the corresponding L-series. In particular, by the root number
w(E/F ) of E over a field F , we will mean the root number of E over F . Similarlyw(E, ρ) is the root number of E twisted by the Artin representation ρ.
Recall from section 4 that PGL2 (Fp ) admits a unique non-trivial 1-dimensional√
representation χ corresponding to the quadratic subfield L = Q( p∗ ) in K fixedby PSL2 (Fp ).
Theorem 9. The root number w(E/Q(C)) of E over Q(C) is +1 if p ≡ 1 (mod 4)and −1 if p ≡ 3 (mod 4). The root number of the twist by the twisted Steinberg
representation w(E, χ ⊗ ρSt ) is equal to −1. Over Q(E[p]) and over K, the root
number is always +1.
p−1

Let s be (−1) 2 . So L is a real quadratic field if and only if s = +1. We list
the root numbers again in the following table.
w(E/Q) = apw(E, χ) = −sw(E/Q(C)) = sw(E/K) = +1

w(E, ρSt ) = ap · sw(E, ρSt ⊗ χ) = −1

We refer to Rohrlich’s article [Roh06] for the computation of the root number of E
twisted by representations of PGL2 (Fp ) in the case E has good reduction at p.
Proof. We simply apply the nice formula of Vladimir Dokchitser in [Dok05]. For
any number field F , his theorem 3 asserts that w(E/F ) = (−1)u+s where u is thenumber of infinite primes in F and s is the number of places where E has split
multiplicative reduction. So we will need the following
Lemma 10. The prime p decomposes in Q(C) as p · qp for two prime ideals p andq with residue field Fp . The reduction is split over these primes if and only if it issplit at p over Q.
Proof. We suppose first that E has split multiplicative reduction at p.
From the description of E as a Tate curve over Qp , we get a short exact sequenceof Gal(Q̄p /Qp )-modules0 µ[p]

 E[p]

 Z/pZ 0.

This sequence is not split. Hence there is only one cyclic subgroup C of order p
on E defined over Qp . The other subgroups C are defined over a field obtained byadjoining a p-th root u of the Tate parameter qE to Qp . Since cp = ordp (qE ) 6 5

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is not divisible by p, the extension Qp (u) is a totally ramified extension of degree
p. So there are at least two embeddings of Q(C) into Q̄p , i.e. there are at least twoplaces above p in Q(C) of which one, say p, is non-ramified and the other, say q
has ramification index eq = p. Since the degree of [Q(C) : Q] is equal to p + 1, we
can not have any other places and the residue field degrees are fq = fp = 1. This
proves the claim for curves with split multiplicative reduction.
Suppose now that the reduction is non-split multiplicative. Then there is aquadratic extension M of Q, inert at p, over which the curve acquires split multiplicative reduction. See [Sil94, Ex. V.11]. The only quadratic sub-extension in K is
the one fixed by PSL2 (Fp ) which must be the unique quadratic extension unramified
outside p and therefore ramified at p. Therefore K is linearly disjoint from M .
Hence, we can pass from Q to M over which E has split reduction at the uniqueunramified prime above p. Then the decomposition in the claim holds for M ·Q(C)/M . He can get down to Q(C)/Q afterwards.
In fact one can prove easily that Q(E[p]) has p2 − 1 places above p with e =p · (p − 1) and f = 1. Similarly for K, the PGL2 (Fp )-extension, there are p + 1 places
above p with e = p · (p − 1) and f = 1. Since f is always equal to 1, the type of
reduction at a place above p can never change from non-split to split multiplicative
reduction.
Since p is odd, we know that E(R)[p] = Z/pZ. Therefore we have that E(C)[p]splits into Z/pZ ⊕ µ[p] as a Gal(C/R)-module. Therefore there are exactly two
embeddings of Q(C) into R, i.e. two real places in Q(C) and p−1
2 complex places.
Hence we can compute the various root numbers. For Q(C), this yields u =2 + p−1
2 and s = 2, if the reduction is split, and s = 0, if the reduction is non-split.
Therefore we conclude thatp−1

w(E/Q(C)) = (−1) 2 = s .
For Q(E[p]) there are no real places, so u ≡ 0 (mod 2) and s = p2 − 1 or s = 0.
Finally for K, we have no real places either and p + 1 prime ideals above p.
The part of St on which the action of the complex conjugation is −1 has dimension p−1
2 . The subspace of St fixed by the inertia group Ip at any place abovep, which can be represented as the group of matrices ( ∗0 1∗ ), is 1-dimensional withtrivial action of Frobenius on it. Therefore theorem 1 in [Dok05] implies−

w(E, ρSt ) = w(E/Q)p · (−1)dim(ρ ) · (−ap )dim ρ−dim ρ
p−1

p−1

Ip

· det(Frobp |ρIp )= app · (−1) 2 · (−ap )p−1 · 1 = (−1) 2 · ap = s · ap .

It is also possible to use the previous computations of w(E/Q(C)) and the fact
Q(C)that IndQ= ⊕ ρSt together with the Artin formalism. With the same sort ofcomputation one finds the versions twisted by χ.
From the above theorem and the parity conjecture, it is natural to expect thatthe rank of the Mordell-Weil group E(Q(C)) is even if and only if p is congruent

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Self-points on elliptic curves of prime conductor13

to 1 modulo 4. We can only prove a weaker result by replacing the rank of theMordell-Weil group by the corank of the Selmer group.
For any number field F , let X(E/F ) be the Tate-Shafarevich group of E overF . We will denote by Selp∞ (E/F ) the usual p-power Selmer group which fits into
the exact sequence0 E(F ) ⊗ Qp/Zp

 Selp∞ (E/F ) X(E/F )(p) 0.

By the corank of the p-power Selmer, we mean the rank of the Pontryagin dualof Selp∞ (E/F ). Hence if the p-primary part X(E/F )(p) of the Tate-Shafarevichgroup is finite, then this corank is nothing else but the rank of the Mordell-Weilgroup E(F ).
Theorem 11. Let E be an elliptic curve of conductor p and let Q(C) be the fieldobtained by adjoining a cyclic group of order p in E to Q. Then the corank of the
p-power Selmer group is even if p ≡ 1 (mod 4) and odd if p ≡ 3 (mod 4).
Proof. One way to state the parity conjecture is the say that the parity of the
p-power Selmer group is even if and only if the root number is +1. In our when caseE admits an isogeny of degree p over Q(C), the parity conjecture has been provedin [DD06]. Hence the theorem follows from theorem 9.
Equally well one can also show the theorem directly. The proof of Shuter
in [Shu06] in the case of good reduction can be adapted to our situation. The
result actually does not differ from Shuter’s result.

7.1. Examples
In the following examples we will assume that all relevant Tate-Shafarevich groups
are finite. For the curves of conductor 11, the above theorem shows that the rankover Q(C) must be odd. If we even believe that the points constructed via isogenies
in 5 are linearly independent of the self-points then the rank would have to be atleast 3.
On the other hand, the curve of conductor 17 must have even rank over Q(C).
Since the self-point PC is of infinite order, the rank has to be at least 2.
For the curve of conductor 43, the rank over Q(C) should be odd. The curve hasa point of infinite order in E(Q) and there is the self-point PC which is of infinite
order. Hence the rank of E(Q(C)) has to be at least 3.
In the following table, we summarise the situation. The first line contains theroot number w(E/Q) = ap , the second the value of s and the third the number of
isogenous curves of E over Q. Finally the last line gives a lower bound on the rankof E(Q(C)) assuming that the Tate-Shafarevich groups are finite and that points
constructed via isogenous curves are linearly independent.

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Curve
ap
s
i
r

11a1
+1
−1
3
3

17a1
+1
+1
4
4

19a1
+1
−1
3
3

37a1
−1
+1
1
2

37b1
+1
+1
3
4

43a1
−1
−1
1
3

53a1
−1
+1
1
2

61a1
−1
+1
1
2

67a1
+1
−1
1
1

73a1
+1
+1
2
2

8. Numerical examples
We use the analytic point of view of the modular parametrisation in order to givean experimental and a computational study of our self-points. Let H denote theupper half plane. The space X0 (p) can be defined as the quotient of H ∪ P1 (Q) by
the congruences subgroup Γ0 (N ). The modular parametrisation is then given by
the following diagram.
ϕE

X0 (p)φE E(C)
℘
C/ΛE

where ΛE = Zω1 ⊕ Zω2 is the period lattice attached to E with ω1 the real and ω2
the imaginary period of E. The map ℘ is the analytic isomorphism from C/ΛE to
E(C) given by the Weierstrass function and its derivative. For τ ∈ X0 (N ) \ {cusps},
the value φE (τ ) is given by the converging series
φE : X0 (p)
τ

 C/Λ
X an
qn

n

with q = exp(2iπτ ) .

n>1

This is a rapidly converging series; bounding the coefficients an /n by 2 it is easy thocontrol the error made by truncating the series. One can also compute efficiently
φE (0) since√X an
φE (0) = L(E, 1) = 2
e−2πn/ N ∈ C/Λ .
nn>1

As above the error made by truncating the series of the right hand size can be easily
controlled. Since we know that ϕE (0) is a rational torsion point of order dividing
the numerator T of (p − 1)/12 in E(Q) we just have to recognise the number φE (0)in the finite set {jω1 /T, j = 0, 1, . . . , T − 1}. This can be done as soon as the error
term is smaller than ω1 /2T .
The index of the congruences subgroup Γ0 (p) in SL(2, Z) is p + 1 and as a set ofrepresentative of SL2 (Z) modulo Γ0 (p) we choose the matrices
M0 = ( 10 01 )Mj = ( 01 −1
j )for j = 1, 2 . . . , p .

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With our analytic point of view, the self-points xC = (E, C) where C runs throughall cyclic subgroups of order p are the p+1 points τ0 = M0 τ , τ1 = M1 τ ,. . . ,τp = Mp τ
where τ = ω2 /ω1 . Our goal is to write down the points PC = ϕE (xC ) as an algebraic
point in E(Q). For this, consider the polynomialsY
Ax (X) =(X − x(ϕE (τj )))
06j6p

Ay (X) =Y

06j6p(X − y(ϕE (τj )))in Q[X] where x(.) and y(.) are the x and the y-coordinate functions of the minimal model of E. Since Q(C) is a primitive extension, the polynomials Ax and Ay
are irreducible over Q. In our numerical applications, those polynomials are given
by real approximation of their coefficients, in order to recognise them as rational
polynomials one need to bound their denominators.
Proposition 12. Let µ = cp if ϕE (0) 6= O and µ = (p + 1)cp otherwise. Then thepolynomials p2µ · Ax (X) and p3µ · Ay (X) have p-integral coefficients.
b
Proof. For a point P 6= O in the formal group E(m)with m the maximal ideal in1
the integer ring of Q̄p , we call − 2 logp |x(P )|p ∈ Q the p-adic valuation of P wherelogp is the logarithm in base p and | · |p is the normalized p-adic absolute value. All
points outside the formal group are said to have p-adic valuation 0.
e corresponds to the point xCe close to the cusp ∞, the exact denominator is
If C
given in the proof of Proposition 3 as we know that ϕE (xCe ) ∈ Ê(pcp Zp ). So PCe has
p-adic valuation cp . Each of the other p points xC is defined over a totally ramified
extensions of degree p of Qp . All the points xC are conjugate to each other, so they
have the same p-adic valuation say λ ∈ p1 Z. Proposition 4 asserts thatX
ϕE (xC ) = ϕE (0) + ϕE (xCe ) .
e
C6=CThe sum on the left hand side must have p-adic valuation greater or equal to λ. If
ϕE (0) 6= O then the right hand size of the equation above has p-adic valuation 0
and so λ = 0. Whenever ϕE (0) = O then the right hand size has p-adic valuation
cp and so λ 6 cp .
We believe that we always have λ = 0.
Proposition 13. Suppose that the genus of X0 (p) is 1 or that E is an involutary
curve (i.e. E ' X0 (p)/wp , where wp is the Fricke involution), then the polynomials
p2cp · Ax (X) and p3cp · Ay (X) have integral coefficients.
Proof. Suppose that we are in the first case, the curve is the strong Weil curve andthe genus is 1 hence the modular parametrisation is an isomorphism. We continue

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Christophe Delaunay, Christian Wuthrich

to write X0 (p) for the minimal model over Z. Let υ be any place in Q(C) above` 6= p. Since the curve E has good reduction at υ, the point xC can not belong to
the kernel of reduction
X0 (Q(C)υ )^X
0 (p)(Fυ )where Q(C)υ is the completion of Q(C) at υ and Fυ the reduction. Since E isisomorphic to X0 (p), we see that the denominator ideal of x(PC ) must be prime toυ. For the prime p, we are in the first case of proposition 12 since ϕE (0) is different
from O for all curves with X0 (p) having genus 1.
If E is involutary then we are in one of the following nine cases:
E = 37a1, 43a1, 53a1, 61a1, 79a1, 83a1, 89a1, 101a1 or 131a1.
We can use the same argument for any place υ not above p in Q(C). Now, we have
ϕE (0) = O and we know that p2(p+1)cp · Ax (X) have integral coefficients hence we
can compute it exactly and since there are finitely many curves we can see that infact p2cp · Ax (X) ∈ Q[X] in all cases.
Whenever the genus of X0 (p) is not 1 and E is not involutary, the proposition
above is false in general. Numerical experimentions show that other primes than p
could appear in the denominator of Ax (X) and Ay (X). Those “extra” primes canbe large and are not well understood. For example, we computed numerically thepolynomial Ax (X) for the curve E = 37b1. We obtain a polynomial in Q[X] which
agrees with Ax (X) up to a large precision but we are not able to prove that it is
the correct one (nevertheless a finite number of tedious computations could do it).
We found that the denominator of Ax (X) should be
32 · 72 · 376 · 41371792 · 948438373827592 .
The factor 376 is explained by Proposition 12. The occurrence of the other factors
is rather surprising (especially for the large ones) and it would be interesting tounderstand where they come from.
When the genus is 1 or the curve E is involutary, we make use of the Proposition
13 to write down the point PC = ϕE (xC ) as an algebraic point. We explain this by
the following examples.
8.1. Example N = 11
ConsiderE = 11a1 : y 2 + y = x3 − x2 − 10 x − 20

11a2 : y 2 + y = x3 − x2 − 7820 − 263580
11a3 : y 2 + y = x3 − x2

the three elliptic curves, up to isomorphism, with conductor N = 11 given in Cremona’s table [Cre97]. The curve E =11a1 is the strong Weil curve. We use theFebruary 22, 2007 9:36 WSPC/INSTRUCTION FILEjsp

Self-points on elliptic curves of prime conductor

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system pari-gp [PAR06] to perform the following computations. We have L(E, 1) =
ω1 /5 and then the corresponding point in E(C) is ϕE (0) = ℘(ω1 /5) = (16, −61)
which is, of course, a rational point of order 5. We put τ = ω2 /ω1 ≈ 1/2 + 1.1494ı
and obtain


10 
Y
−1
X − ϕEAx (X) = (X − ϕE (τ ))τ +j
j=0≈ X 12 − 1858429.3660X 11 + · · · + 1027552848072306586730.7357
By the proposition above, we easily recognise
Ax (X) = 11−10 (25937424601 X 12 −48202871557476252 X 11 −627575688471844224310 X 10−4586587322380883649178756 X 9 +189861110415625174383936023 X 8−9450307537215069858510760088 X 7 −13176917774298176001346511796 X 6+558725269921007151668021541368 X 5 +4102246055136279443069069843703 X 4+12745046520678761793965586279924 X 3 +25688385347866823866818727620042 X 2+32963462539585829067954484001996 X+26652074520418260289484123648825)

This polynomial defines the primitive number field Q(C) of degree 12. The following
polynomial A(X) defines an isomorphic number field
X 12 −4 X 11+55 X 9−165 X 8+264 X 7−341 X 6+330 X 5−165 X 4−55 X 3+99 X 2−41 X−111
One can show that the class group of Q(C) is trivial. One can perform the same
computations for the y-coordinate function. We find the following algebraic point
P1 = PC = (x1 , y1 ) withx1 = 2−1 ·5−3 ·11−10 ·19−1 ·(133792802077952089 θ 11 −312848945005283368 θ 10 −502878903201648831 θ 9+6475439902255323868 θ 8 −11358894741986615604 θ 7 +17292758289068725628 θ 6−18719462641364369973 θ 5 +16016249446153991254 θ 4 +982764516960529358 θ 3−2408156353187544234 θ 2 +8344249326459947483 θ+7914557261562811262)

y1 = 2−1 ·5−3 ·11−15 ·19−1 ·(28765696339563795386130989 θ 11 −65657747550440506261377918 θ 10−112696985861148667470565931 θ 9 +1388431928553041563545929168 θ 8 −2362133632267562166352755504 θ 7+3540644107632326208066046428 θ 6 −3732014744849429143581644773 θ 5 +3085273426989748502235208404 θ 4+556569952015915009535873158 θ 3 −638947781627567488171165884 θ 2 +1776899022233271387870124783 θ+1848199666743612190587863962)

where θ is a root of A(X). Note that the numerator of these numbers are actuallydivisible in the ring of integers of Q(C) by the integers different from 11 appearing
in the denominator above; just as predicted by the proof of Proposition 13. As θ isrunning through the roots of A(X) in a fixed Galois closure of Q, the points givenby the above formula describe the conjugates of the image by ϕE of our self-point.
In other words we obtain all the points PC as C runs through the cyclic subgroups

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Christophe Delaunay, Christian WuthrichPof order p in E[p]. Then, one can explicitly compute the point C PC and we findthe 5-torsion point (16, −61) as predicted by Proposition 4.
In fact, one has the easier expression for P1 since we have P1 = (x01 , y10 ) +(16, −61) with
x01 = 2−1 ·5−3 ·11−2 ·19−1 ·(−1669θ11 +33828θ10 −81349θ9 −183828θ8 +1717484θ7 −3471788θ6 +4165033θ5−4576634θ 4 +2336882θ 3 +3088214θ 2 −4748743θ+4861598) ,y10 = 2−1 ·5−3 ·11−3 ·19−1 ·(−233091θ11 +1788042θ10 −2346611θ9 −16073792θ8 +82949376θ7 −148123732θ 6+181279787θ 5 −175063476θ 4 +56212398θ 3 +138667596θ 2 −195077977θ+186910572) .

We can also compute with the same method the algebraic points on E coming fromthe self-points of the isogenous curves 11a2 and 11a3. We find the following points
P2 = (x2 , y2 ) + (16, −61) and P3 = (x3 , y3 ) withx2 = 2−1 ·5−3 ·11−2 ·19−1 ·(1594026θ11 −6388637θ10 +702246θ9 +86307137θ8 −265102036θ 7 +450584852θ 6−609415332θ 5 +678510761θ 4 −389518078θ 3 −208816706θ 2 +158121422θ−162620467) ,y2 = 2−1 ·5−3 ·11−2 ·19−1 ·(−16692116θ 11 +95324467θ10 −78704636θ9 −1037899217θ 8 +4171219776θ 7−7057412532θ 6 +8919208962θ 5 −12272284751θ 4 +9718545598θ 3 −6416053004θ 2 +8604323348θ−356847003)

andx3 = 2−1 ·5−3 ·11−2 ·(11θ11 +40293θ10 −485694θ9 +1502457θ8 −2828496θ7 +4716822θ6 −5419227θ5 +2855721θ4+1781967θ 3 −2734116θ 2 +1252592θ+1304413) ,y3 = 2−1 ·5−3 ·11−2 ·(−9727989θ11 +90416293θ10 −386035694θ 9 +853672457θ 8 −1247148496θ 7 +1474894322θ 6−1275534227θ 5 +521439721θ 4 +268493467θ 3 −363165616θ 2 +55053592θ+386303413) .

Using the system magma [BCP97], we can compute the canonical Néron-Tate heightof these points.
ĥ(P1 ) = 3.733

ĥ(P2 ) = 6.420ĥ(P3 ) = 2.117

The determinant of the height matrix is equal to 1301.155. This proves the assertion
that the the three selfpoints P1 , P2 and P3 generate a group of rank 3 in E(Q(C)).
From the proof of Theorem 8, we can now deduce that the rank of the group
generated by higher self-points in E(Kn ) is at least
rank E(Kn ) > 3 · pn + 3 · pn−1 − 3 .
8.2. Conductor 17 and 19
Let E be the curve 17a1; its isogeny class contains four elliptic curves. The field
Q(C) is defined by the polynomial
X 18 −7X 17 +17X 16 +17X 15 −935X 14 +799X 13 +9231X 12 −41463X 11 +192780X 10 +291686X 9 −390014X 8+6132223X 7 −3955645X 6 +2916112X 5 +45030739X 4 −94452714X 3 +184016925X 2 −141466230X+113422599 .

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The four selfpoints P1 , P2 , P3 and P4 coming from the four isogenous elliptic curves
have rather complicated coordinates but can be computed with the same method
as for p = 11. The canonical Néron-Tate height of these points are as follows.
ĥ(P1 ) = 2.707ĥ(P2 ) = 2.271ĥ(P3 ) = 2.896ĥ(P4 ) = 1.786

The determinant of the height matrix is equal to 4594.647. This proves the assertion
that the four selfpoints P1 , P2 , P3 and P4 generate a group of rank 4 in E(Q(C)).
From the proof of Theorem 8, we can now deduce that that the rank of the group
generated by higher self-points in E(Kn ) is at least
rank E(Kn ) > 4 · pn + 4 · pn−1 − 4 .
Let E be the curve 19a1; there are three elliptic curves in its isogeny classes. The
field Q(C) is defined by the polynomial
X 20 −5X 19 +76X 18 −247X 17 +1197X 16 −8474X 15 +15561X 14 −112347X 13 +325793X 12 −787322X 11+3851661X 10 −5756183X 9 +20865344X 8 −48001353X 7 +45895165X 6 −245996344X 5+8889264X 4 −588303992X 3 −54940704X 2 −538817408X+31141888 .

In this case, the canonical Néron-Tate height of the points P1 , P2 and P3 areĥ(P1 ) = 2.257

ĥ(P2 ) = 3.207ĥ(P3 ) = 1.576

The determinant of the height matrix is equal to 469.791. This proves thatrank E(Kn ) > 3 · pn + 3 · pn−1 − 3 .
8.3. Example of the involutary curve E=37a1
Let E be the involutary curve 37a1; there is just E in its isogeny classes. The field
Q(C) is defined by
X 38 −12X 37 +6845X 34 −4107X 33 +611943X 32 +6419241X 31 −23619357X 30 −1139989573X 29 −7189689558X 28+15195444123X 27 +421676681701X 26 +1049261700469X 25 −24717066801390X 24 −348296732228468X 23−2419574069703120X 22 −8165136099970176X 21 +14475033463029762X 20 +268023003210734612X 19+329661477773764104X 18 −11420015354801245670X 17 −94568058590056572726X 16 −270708469237425691308X 15+701411470253839139591X 14 +8568002617280552745928X 13 +22563667968332689606038X 12−66882215649968841310916X 11 −706898153127401380189661X 10 −2172210500846597000641917X 9−93155470637659787671857X 8 +25476395590222338315403899X 7 +115054283689855001616765285X 6+297976894378693703291782499X 5 +526572254394503227631311356X 4 +665351270233256888369987865X 3+573638845689345417051088091X 2 +357303812435373401026408215X+585578575222824132475605000 .

In this case, the curve E has rank one over Q and hence
rank E(Kn ) > pn + pn−1 .

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Acknowledgments
The authors would like to thank John Coates, Henri Darmon, Vladimir Dokchitser,
Bas Edixhoven, Yoshitaka Hachimori, Xavier-François Roblot, and Mike Shuter for
helpful discussions. The second author is supported by a fellowship of the Swiss
National Science Foundation
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